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Theorem invghm 15455
 Description: The inversion map is a group automorphism if and only if the group is abelian. (In general it is only a group homomorphism into the opposite group, but in an abelian group the opposite group coincides with the group itself.) (Contributed by Mario Carneiro, 4-May-2015.)
Hypotheses
Ref Expression
invghm.b
invghm.m
Assertion
Ref Expression
invghm

Proof of Theorem invghm
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invghm.b . . 3
2 eqid 2438 . . 3
3 ablgrp 15419 . . 3
4 invghm.m . . . . 5
51, 4grpinvf 14851 . . . 4
63, 5syl 16 . . 3
71, 2, 4ablinvadd 15436 . . . 4
873expb 1155 . . 3
91, 1, 2, 2, 3, 3, 6, 8isghmd 15017 . 2
10 ghmgrp1 15010 . . 3
1110adantr 453 . . . . . . . 8
12 simprr 735 . . . . . . . 8
13 simprl 734 . . . . . . . 8
141, 2, 4grpinvadd 14869 . . . . . . . 8
1511, 12, 13, 14syl3anc 1185 . . . . . . 7
1615fveq2d 5734 . . . . . 6
17 simpl 445 . . . . . . 7
181, 4grpinvcl 14852 . . . . . . . 8
1911, 13, 18syl2anc 644 . . . . . . 7
201, 4grpinvcl 14852 . . . . . . . 8
2111, 12, 20syl2anc 644 . . . . . . 7
221, 2, 2ghmlin 15013 . . . . . . 7
2317, 19, 21, 22syl3anc 1185 . . . . . 6
241, 4grpinvinv 14860 . . . . . . . 8
2511, 13, 24syl2anc 644 . . . . . . 7
261, 4grpinvinv 14860 . . . . . . . 8
2711, 12, 26syl2anc 644 . . . . . . 7
2825, 27oveq12d 6101 . . . . . 6
2916, 23, 283eqtrd 2474 . . . . 5
301, 2grpcl 14820 . . . . . . 7
3111, 12, 13, 30syl3anc 1185 . . . . . 6
321, 4grpinvinv 14860 . . . . . 6
3311, 31, 32syl2anc 644 . . . . 5
3429, 33eqtr3d 2472 . . . 4
3534ralrimivva 2800 . . 3
361, 2isabl2 15422 . . 3
3710, 35, 36sylanbrc 647 . 2
389, 37impbii 182 1
 Colors of variables: wff set class Syntax hints:   wb 178   wa 360   wceq 1653   wcel 1726  wral 2707  wf 5452  cfv 5456  (class class class)co 6083  cbs 13471   cplusg 13531  cgrp 14687  cminusg 14688   cghm 15005  cabel 15415 This theorem is referenced by:  gsuminv  15543  invlmhm  16120  tsmsinv  18179 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rmo 2715  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-riota 6551  df-0g 13729  df-mnd 14692  df-grp 14814  df-minusg 14815  df-ghm 15006  df-cmn 15416  df-abl 15417
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